January 23, 2004

**Abstract. **Over the last 20 years, knot theorists have been
extremely good at borrowing ideas from other fields. We've borrowed
from Mathematical Physics and borrowed from Algebra and we have a
Beautiful Theory of Knot Invariants that can claim deep heritage on
either side. But we haven't been so good at returning. While not
entirely impossible, is remains difficult to point at developments in
quantum field theory or quantum algebra (our lenders) that owe
something to our Beautiful Theory of Knot Invariants.

Came Khovanov in 1999 and changed the picture dramatically by
offering Mathematical Physics and Algebra the most valuable prize known
to mathematicians - a *challenge*. For none of them can yet
explain whence comes his "Categorification of the Jones Polynomial" - a
far reaching generalization of the most celebrated member of our
Beautiful Theory of Knot Invariants. The Mathematical Physics and
Algebra underlying the Jones polynomial are deep and substantial, and
there are all reasons to believe that a successful resolution of
Khovanov's challenge will be the same.

In my talk I will quickly describe the Jones polynomial (it's so easy) and then move on to describe Khovanov's homological generalization thereof.

(Khovanov's homology is also a stronger invariant than the Jones polynomial and it is "functorial" in some 4-dimensional sense).

Handout side 1:
QRG.pdf.

Handout side 2:
NewHandout-1.pdf.